Collective behavior, from murmurations to synchronized beating of heart cells, governs some of the most beautiful and important aspects of nature. Likewise, cooperation—the act of sacrificing personal benefits for the common good—is one of the pillars of social evolution, and it is the basis for the emergence of collective organized actions from single-cell organisms to modern human societies. Here we merge these two phenomena into a single model, considering an ensemble of networked oscillators, where each oscillator can be either a cooperator or a defector, and with only cooperators contributing to synchrony. At the same time, the value of the order parameter in the neighborhood of each oscillator is considered as an effective local temperature which determines the strategy updating procedure in the evolutionary game. The emergence of cooperation is thus intertwined with that of synchronization, producing a novel and fascinating dynamics which includes a double explosive transition.

从杂音到心脏细胞同步跳动，集体行为支配着自然界中一些最美丽，最重要的方面。同样，合作（为公共利益牺牲个人利益的行为）是社会进化的支柱之一，并且是从单细胞生物到现代人类社会集体组织行动出现的基础。在这里，我们将这两种现象合并为一个模型，其中考虑了一组网络振荡器，其中每个振荡器可以是合作者或叛逃者，并且只有合作者可以促进同步。同时，每个振荡器附近的阶次参数值被认为是有效的局部温度，它决定了进化博弈中的策略更新过程。